It's currently not possible to automatically rename fields. The configuration isn't changed when a new schema is added to Studio. But, a way around this is to follow the steps below.
Rename component fields
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Edit a schema in the Components area containing both the old and the new field names
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Manually copy the value from the old setting to the new one in the page builder
You can see a list of all pages using your Frontend component in the Components area of the Studio as shown below:
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You can click on the name of where it's used and be taken directly to the page builder.
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After editing and saving all pages, the code using the new field can be deployed
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Edit the schema to only contain the new field name